"heuristic theory of phase transitions" [7], and the strong form of the Gibbs Phase. Rule that it implies, are violated in spaces of long-range interactions.
Communications in Commun. Math. Phys. 106, 459^66 (1986)
Mathematical
Physics
© Springer-Verlag 1986
Generic Triviality of Phase Diagrams in Spaces of Long-Range Interactions Robert B. Israel* Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T1Y4
Abstract. We show that interactions with multiple translation-invariant equilibrium states form a very "thin" set in spaces of long-range interactions of classical or quantum lattice systems. For example, generic finite-dimensional subspaces do not intersect this set. This constitutes a severe violation of the Gibbs Phase Rule.
1. Introduction The work of Daniels and van Enter [1-3] has shown that phase transitions are less stable under long-range perturbations than had been believed, so that Ruelle's "heuristic theory of phase transitions" [7], and the strong form of the Gibbs Phase Rule that it implies, are violated in spaces of long-range interactions. In this paper we extend and generalize those results, and show that the instability of phase transitions is a generic phenomenon, in the sense of Baire category, in these spaces of interactions. This means that phase transitions occur only on a set which is in a sense very "thin," violating much weaker versions of the phase rule. We will consider either a classical or quantum lattice system on 7Lά (see [4] for notation). In the classical case the configuration space at each site is assumed to be a compact metric space. The Banach space Sig consists of those translationinvariant interactions Φ with OeX
(l.i)
where g is some function on the positive integers, and \X\ is the cardinality of X. In order to define equilibrium states by the variational principle, we require g(n) ^ ί/n [in the case g(ή) = ί/n we will write Sig as 0β~\. Stronger conditions on g [e.g. an g(n)^e for some α>0] allow the use of DLR equations, KMS conditions, etc. Our results hold in any of these spaces, and thus are not connected to the •• Research supported by NSERC grant A-4015
460
R. B. Israel
pathologies found in [5] for 38. On the other hand, they do depend on the interactions being allowed to be "long-range" in the sense that the weighting g(\X\) depends only on the cardinality of X and not on its diameter. In particular, a twobody Ising interaction is in any Stg provided it is summable. We are concerned with the basic question, "What does a typical phase diagram look like?" According to the Gibbs Phase Rule, an ^-dimensional subspace of interactions should typically have an n — ί -dimensional set of two-phase coexistence (i.e. where there are two extremal invariant equilibrium states), an n — 2dimensional set of three-phase coexistence, etc. Moreover, one would expect these sets to be manifolds (perhaps with boundary). While these principles may be valid in spaces of short-range interactions, our results show an entirely different picture in 33g: a finite-dimensional subspace which is "typical" in the sense of Baire category will have no points with multiple phases. One may also start out with a given finite-dimensional subspace S of interactions and perturb it, i.e. consider S + M, where M is a finite-dimensional subspace. Our result shows that "typically" the only multiple-phase points in S + M are those in S itself. Actually, the result is even stronger than this: S may be any σ-compact set, i.e. the union of a countable collection of compact sets. In the quantum case, or the classical discrete-spin case, S may be any Banach space of interactions with a norm l|φ||,,*Ξ Σg(\x\)KX)\\Φ(X)L
(1.2)
OeX
where h(X) -> oo as diam (X) -> oo. (To show this is σ-compact, note that its unit ball can be approximated by bounded subsets of finite-dimensional subspaces, and therefore is totally bounded.) So we can conclude that any "typical" long-range perturbation destroys the long-range order of all short-range interactions. There are a number of possible ways to defend the physical Gibbs Phase Rule from the results of this paper. One might say (1) Physics is not generic, or (2) Physical interactions are short-ranged. These objections are related, and both have some merit. Regarding (1), we might remark that the fact that a property of the members of some space is "generic" does not mean we should expect any particular member to have that property, but instead says something about the sort of additional conditions that might ensure that it does not have the property. In this case, it would seem that the only reasonable way to save the Gibbs Phase Rule is to insist on short-ranged interactions. On the other hand, there are indeed long-range forces in nature, such as the magnetic forces that occur in real ferromagnets. Admittedly, the magnetic dipoledipole interaction does not fit in any of our Banach spaces, but its effects are in a way analogous to those of the perturbing interactions of [1-3] and Theorem 1: while negligible on the atomic scale in comparison to the exchange interaction, on a larger scale it causes the formation of "domains" which interfere with true "ferromagnetic" long-range order. The energy-minimization considerations that govern the formation of magnetic domains have some resemblance to the arguments of these papers. So perhaps our results are not completely devoid of physical significance.
Triviality of Phase Diagrams
461
In Sect. 2 we basically generalize the results of Daniels and van Enter [1-3] on existence of perturbations that destroy long-range order. The main new features are the following: (1) symmetry is not required, so that we can deal with all interactions and observables; (2) the conclusions hold for all invariant equilibrium states, not just those that are extremal Gibbs states; (3) we note that we can perturb not just a single interaction Φ, but a large class of them. Then in Sect. 3 we combine the existence result with topological arguments to prove the main result (Theorem 2) of this paper. As an application, we prove a result similar to a theorem of Sokal [8]. A closely related result in a more general context of affine continuous functions on a Choquet simplex is Theorem 3.2 of [6]. That theorem includes Theorem 2 of this paper for the space J*.
2. Existence of Order-Destroying Perturbations For Φ e @ let JN(Φ)=
Σ \\Φ(X)\\ \X\-ιmin(l,diam(X)/JV).
(2.1)
OeX
Note that the sum converges, and JN(Φ)^0 as JV->oo for any Φ. An interaction Φ may be considered as "short-ranged" if JN(Φ)-+0 relatively quickly as JV->oo. Given any sequence fN, we denote by Uf the set of interactions Φ such that JN(Φ)0 and h(X)->oo as diam(X)-χx). The following lemma relates membership in this space to decay of JN(Φ). Lemma 1. For any h(X)>0 with h(X)-*oo as diam(X)-κx) there is a sequence fN->0 such that ^hQ Uf. Conversely, for any sequence fN>0with fN^0 as N-^co, Uf is contained in some Banach space ^h. Proof. Given h, let kN{x)
-
We have kN(X)^\/h{X)^ as diam(X)^oo for each AT, while monotonically as iV-> oo for each X, so it is easy to see that /^(JQ-^O uniformly in X as N->oo. Let fN^0 with maxkN(X)/fN^0 as ΛΓ-xx). Now if Φe%h, then x JN(Φ)^
Σ
\ \ Φ ( X ) \ \
OeX
N^l9 2ifNj^0
as ;-*oo,
(2.7)
2i
2 /Nj-+0 as 7-^00, (2.8) and let m^JV^. Note that | | 5 P | | ^ M|| 2 ^(2|7|). Now if ρί and ρ2 are invariant states with \ρί(A) — ρ2(A)\'^ε, it is easy to show that ρ = (ρ1+ρ2)β satisfies + ε2/4.
(2.9)
Of course iϊρί and ρ2 are equilibrium states for an interaction Φ + tΨ, the same is true for ρ. To show that ρ can not be an equilibrium state for this interaction, we will find another invariant state ρ such that s(ρ) - ρ(Aφ + tΨ) > s(ρ) - ρ(Aφ+tΨ), and that will prove the theorem.
(2.10)
Triviality of Phase Diagrams
463
We will obtain ρ from ρ by "decoupling" large blocks of the lattice. Given the invariant state ρ and a positive integer iV, define ρN as follows: cover the lattice by disjoint cubes of side JV (which we will call "blocks"). Let ρN be the product of the restrictions of ρ to each block (in probabilistic terminology, we make the blocks independent). Let ρN be the average of translations of ρN by all elements of a block. Then ρN is an invariant state. We wish to prove (2.10) with ρ = ρNk for some k. Note that s(ρN)^s(ρ), so (using linearity) it suffices to prove ρ(Λφ) + tρ(ΛΨ) > ρNk(Aφ) + tρNk(AΨ).
(2.11)
The idea is that the difference between ρ and ρNk on Aφ or on ΛΨ. for j k we get a "volume effect" which will be dominant if k is sufficiently large. The following estimates will make that precise. First, there is a constant Kγ (independent of ρ and N) such that if B e stfx, then K,\\B\\ dmm(X)/N,
(2.12)
since ρN(τxB) — ρ(B) = 0 unless x + X intersects more than one block (this is the "surface effect"). This implies that if Φ e Uf N
(2.13)
for N sufficiently large. Moreover, there is a constant K2 (depending only on A and X) such that Since Nk>mj
\(§N-ρ)(AΨJ
= \(ρN-ρ)(cm(A)2)\
